Initial value representation

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3.8.13 (and/or the equivalent time integral of the flux-flux correlation fiinction). While this approach has to date only been applied to problems with simplified hannonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a fiirther classical or quasiclassical approximation. 

Sun X, Wang H B and Miller W H 1998 Semiclassical theory of electronically nonadiabatic dynamics Results of a linearized approximation to the initial value representation J. Chem. Phys. 109 7064... 

Due to the development of efficient initial-value representations of the semiclassical propagator, recently there has been considerable progress in the semiclassical description of multidimensional quantum processes [104—111,... 

Considering the semiclassical description of nonadiabatic dynamics, only the mapping approach [99, 100] and the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller [112] appear to be amenable to a numerical treatment via an initial-value representation [114, 116, 117, 121, 122]. Other semiclassical formulations such as Pechukas path-integral formulation [45] and the various connection... 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

The breakdown of the SH scheme in the case of classically forbidden electronic transitions should not come as a surprise, but is a consequence of the rather simplifying assumptions [i.e., Eqs. (37) and (43)] underlying the SH model. On a semiclassical level, classically forbidden transitions may approximately be described within an initial-value representation (see Section VIII) or by introducing complex-valued trajectories [55]. On the quasi-classical... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

Following a brief introduction of the basic concepts of semiclassical dynamics, in particular of the semiclassical propagator and its initial value representation, we discuss in this section the application of the semiclassical mapping approach to nonadiabatic dynamics. Based on numerical results for the... 

As a consequence, the semiclassical propagator is given as a phase-space integral over the initial conditions qo and Po, which is amenable to a Monte Carlo evaluation. For this reason, semiclassical initial-value representations are regarded as the key to the application of semiclassical methods to multidimensional systems. 

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

W. H. Miller. The semiclassical initial value representation Adding quantum effects to classical molecular dynamics simulations. J. Phys. Chem. A, 105 2942, 2001. 

X. Sun and W.H. Miller. Semiclassical initial value representation for electronically nonadiabatic molecular dynamics. J. Chem. Phys., 106 6346, 1997. 

Using the Semielassieal Initial Value Representation to add Quantum Effeets to Classieal Moleeular Dynamics Simulations Nikitin E.E. 

X. Sun, H. B. Wang, and W. H. Miller (1998) On the semiclassical description of quantum coherence in thermal rate constants. J. Chem,. Phys. 109, p. 4190 X. Sun, H. B. Wang, and W. H. Miller (1998) Semiclassical theory of electronically nonadaibatic molecular dynamics Results of a linearized approximation to the initial value representation. J. Chem. Phys. 109, p. 7064... 

W. H. MiUer (2002) An alternate derivation of the herman-kluk (coherent state) semiclassical initial value representation of the time evolution operator. Mol. Phys. 100, p. 397... 

---